In some diatomic molecules, the force each atom exerts on the other can be approximated by F = − C / r 2 + D / r 3 , where r is the atomic separation and C and D are positive constants, ( a ) Graph F vs. r from r = 0.8 D/C to r = 4 D / C . ( b ) Show that equilibrium occurs at r = r 0 = D/C. ( c ) Let Δ r = r – r 0 be a small displacement from equilibrium, where Δ r ≪ r 0 . Show that for such small displacements, the motion is approximately simple harmonic, and ( d ) determine the force constant. ( e ) What is the period of such motion? [ Hint : Assume one atom is kept at rest.]
In some diatomic molecules, the force each atom exerts on the other can be approximated by F = − C / r 2 + D / r 3 , where r is the atomic separation and C and D are positive constants, ( a ) Graph F vs. r from r = 0.8 D/C to r = 4 D / C . ( b ) Show that equilibrium occurs at r = r 0 = D/C. ( c ) Let Δ r = r – r 0 be a small displacement from equilibrium, where Δ r ≪ r 0 . Show that for such small displacements, the motion is approximately simple harmonic, and ( d ) determine the force constant. ( e ) What is the period of such motion? [ Hint : Assume one atom is kept at rest.]
In some diatomic molecules, the force each atom exerts on the other can be approximated by
F
=
−
C
/
r
2
+
D
/
r
3
, where r is the atomic separation and C and D are positive constants, (a) Graph F vs. r from r = 0.8D/C to r = 4D/C. (b) Show that equilibrium occurs at r = r0= D/C. (c) Let Δr = r – r0 be a small displacement from equilibrium, where
Δ
r
≪
r
0
. Show that for such small displacements, the motion is approximately simple harmonic, and (d) determine the force constant. (e) What is the period of such motion? [Hint: Assume one atom is kept at rest.]
of a copper wire of uniform cross section and
Ex. 70: find the energy stored per unit volume
of length 1.5 m, when it is stretched to a length
of a copper wire of uniform cross section and
of length 1.5 m, when it is stretched to
length
a
of 1.51 m by a stress of 3 x 102 N/m2.
The potential energy of two atoms in a diatomic molecule can be approximated by the Lennard-
Jones potential U(r) = a/r¹² — b/r6, where r is the distance between the two atoms, and a and b
are positive constants.
a) Find the force F(r) on one of the atoms as a function of r.
b) Find the equilibrium distance between the two atoms. Is this equilibrium stable?
c) Suppose the distance between the two atoms is equal to the equilibrium distance found in part
b). What minimum energy must be added to the molecule to break the two atoms apart? (This
is called the dissociation energy of the molecule.)
The potential energy of a nitrogen atom in an ammonia (NH3) molecule varies with position as U(x) = x4 - 2ax2 (a is some constant)
a) Give an expression for the force F on the nitrogen atom and sketch F(x).
b) At what positions will the nitrogen atom be in stable equilibrium?
Chapter 14 Solutions
Physics for Scientists and Engineers with Modern Physics
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